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Summary MA261 (Multivariate Calculus) Study Guide $5.49
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Summary MA261 (Multivariate Calculus) Study Guide
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Study guide of the whole MA261 course. Includes Exam 1, Exam 2, and Final Exam topics. Topics include vectors, equations of lines and planes, partial derivatives, integrals in polar, rectangular regions, and spherical, gradients, max/min, mass, vector fields, curl/divergence, and theorems.

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  • November 10, 2024
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  • 2024/2025
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multivariate calculus
-




equations & Plane
O




V ECT OR S
. of lines
· . . . . . .




notation : = =
(x , y z) ,
= xi + yj + zk equation of line : (t) (Xo , Yo zo]
=
,
+ + ·
(x0 , Yoz0)
↑
(x2 + y2 + zz
⑳


magnitude (length) : (t) =
equation of plane : ((X-Xo) (y-yo) (z-zo)) n
, ,
· =
0 ↑
(X0 , yo , z0)



1a VECTOR RULES ↳ QUADraTIC SURFACES
↳

directed rector from Po <Xo , Yo zo) =
,
to Pi [X , Y 1, zi] :
=
·
ellipsoid : 4= ++757/



elliptic paraboloid :
III
i = PP =
Pi Po (X,- = -
X0 , Y1 yo , z -
z0) ·




dot product of a =
(ax My az) and , ,
J =
(bx , by bz7
,
: ·
hyperbolic paraboloid (saddle): -
a -
5 axbx + ayby + azbz
=
·
cone : d
a and 5 hyperboloid of 1 sheet :=
n
·
angle between :



9 6
-
:



cosO hyperboloid of 2 sheets :
-

=
191151 ·




projection of 5 along :

proj = S
cross product in R3 :



i5E
x5 =

ax ay az
(aybz-Azby)5 -
(axbz -

azbx)j + (ax by -
aybx)
by by be
↳ x5 is
normal/orthogonal/perpendicular to both a <5
↳ areas of shapes :
partial derivatives
y) 22f


=
1 lim f(x + h , y) f(x
·
-
,


A= laxol fxy fyx =


ey2X22)
=
V
3
· A
lim f(x , y + h) -
f f(X , y) 22f
; fyy 2y2
=
fxx

-
=
H 2x2

· v =
/ :
(bX)/
VECTOR vALUE FUNCTIONS ↳
equation of tangent plane to graph of z = f(x Y), at (Xo , Yo · zo) :




r(t) = (f(t) , g(t) , h(t)) z -

zo = fx (Xo Yo)(X Xo)
,
-
+ fy(X0 yo) <y Yo)
,
-




*
tangent vector : v'(t) differential of w =
f(X , Y , z) :




unit tangent rector : F(t) = -
dw =
df = + dy + dz
ex

T'(t)
unit normal vector : NCt) =
IT'(t) Linear approximation :
((X, y) =
f(a b) , + fx(a b)(X a) ,
-

+ fy(a b)(y b)
,
-




↳
(a , b)
>
-



L(X , y) = f(X , y) near



integrals : Chain Rule :

Suct)dt JSf(t)dt Sg(t)dt ShCtdt]
ya
=
, ,




arc length : 1= Srct)(d -


IT'(t)
curvature : k Level curves f(x , y z)
·
Ir(t) : k
=
=
,




position , velocity acceleration ,
·
Limits of functions :
lim
(t) = v'(t) =
v "(t) f(X , y , z) DNE if different approaches

Newton's 2nd Law : F ma = to (a , b) yield different limits

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